Multi-chart geometry images

ABSTRACT

Techniques and tools for mesh processing are described. For example, a multi-chart geometry image represents arbitrary surfaces on object models. The multi-chart geometry image is created by resampling a surface onto a regular 2D grid, using a flexible atlas construction to map the surface piecewise onto charts of arbitrary shape. This added flexibility reduces parameterization distortion and thus provides greater geometric fidelity, particularly for shapes with long extremities, high genus, or disconnected components. As another example, zippering creates a watertight surface on reconstructed triangle meshes. The zippering unifies discrete paths of samples along chart boundaries to form the watertight mesh.

TECHNICAL FIELD

The invention relates generally to three-dimensional computer graphics.More specifically, the invention relates to new multi-chart geometryimage representations and processing techniques for arbitrary surfacesof object models.

BACKGROUND

Computer graphics have evolved from the crude data displayed onmonitors, printers or hardcopy plotters of early computers to thecomputationally complex three-dimensional (3D) images of today. Earlycomputers could not show complex computer graphics primarily becausehardware was expensive and computer performance was slow. Since then,advances in technology have caused computer graphics hardware prices todrop dramatically and performance of graphics hardware has drasticallyimproved. At the same time, graphics processing techniques have becomemore sophisticated and efficient. Today, many desktop computers support3D computer graphics through improved and relatively inexpensivegraphics accelerating hardware (accelerators). Video game consoles alsoinclude specialized graphics hardware.

A graphics accelerator is a type of video device that contains its ownprocessor to boost performance levels. These processors are specializedfor computing graphical transformations, so they achieve better resultsthan the general-purpose processor (i.e., the central processing unit,or CPU) used by the computer. In addition, a graphics accelerator freesup the computer's CPU to execute other commands while the graphicsaccelerator is handling graphics computations. One basic function ofgraphics accelerators is displaying graphical primitives (e.g., lines,polygons, curves, and textual characters). Another basic function isputting constraints on primitives, for example, to enforce how graphicalobjects move in relation to other objects, such as how a revolving doorturns around its pillar or how a drawer moves along runners in itscabinet. Other functions include performing algorithms for draggingobjects, rubberbanding (i.e., graphically resizing objects), andtransforming (e.g. rotating, scaling, or translating) an image. Graphicsaccelerators typically also process data structures for storing objectmodels such as geometric models.

I. Geometric Models

Geometric modeling is an important area of 3D computer graphics, whichdeals with the representation and manipulation of graphical objects in acomputer. In early stages, computer graphics were two-dimensional (2D).Displayed objects were defined by complicated and lengthy sequences ofdrawing commands. For performance reasons, it became important todistinguish between the model of the object to be displayed and thedisplay primitives on the screen at a given time for the object. Forobjects viewed from different perspectives, distances, or contexts, itbecame more efficient to represent the object with one model from whichdifferent images were rendered as necessary, rather than as a collectionof different images (with drawing primitives) for various views. To“render” an object means to create an image for display from the object,considering the object itself and potentially other information such aslight sources and perspective. The distinction between the object modeland display primitives became even more important in 3D computergraphics because of the increased complexity of the objects andincreased display detail.

Today, geometric modeling provides a way to model physical and virtual(computer generated) objects on a computer. In their most basicapplication, geometric models are computational (symbolic) structuresthat capture the spatial aspects of the objects of interest. In additionto spatial information, many models store information such as color,texture, and so on to accurately represent objects. These geometricmodels form the foundation of powerful simulation and modeling tools.

For example, through the use of geometric modeling, computers canperform realistic simulations of car crashes in a non-destructivefashion, analyze the stresses in aerospace structures during flight, andperform pre-surgical planning such as simulating the implantation of anartificial limb.

Moreover, 3D computer graphics have become more important in multimediaand Internet applications. For example, movie special effects and videogames are increasingly driven by 3D computer graphics. Online retailerscan now showcase online object models of sales items to customers, whichallow customers to peruse an item before buying it. All of these andother functions use geometric modeling and illustrate its usefulness.

II. Meshed Object Models Overview

The use of polygonal meshes for the representation of highly complexgeometric objects has become the generally accepted standard in mostcomputer graphics applications. The reason for this is that flexible andeffective processing and storage techniques have been developed whichusually work for models and shapes with arbitrary topology.

A mesh is made up of connected points. In one common kind of model, eachof the points has three coordinate values (x, y, z), which correspond towidth, height, and depth. Some meshes are entirely computer generated.Other meshes are based on measured data points, where the raw data isobtained by mechanical or optical scanning of an object's surface.Adjacent points may be connected to produce triangles, quadrilaterals,or other polygonal shapes. For example, points are converted into atriangle mesh by connecting nearby points into triangles according tosome set of rules. Generally, triangle meshes are preferred due to theiralgorithmic simplicity, numerical robustness, and efficient display.Geometrically, a triangle mesh has a piecewise surface (e.g. a surfacecomposed of sub-regions or sub-divisions) consisting of triangularshapes (or “faces”) joined together along their edges. An advantage oftriangle meshes is that they can be used for many stages of the typicalprocessing pipeline in geometric design and graphics applicationswithout the need for data conversion. This decreases the overallprocessing time and reduces the potential for round-off errors (whichare more likely to be introduced with more data conversions).

A regular mesh has uniform patches and a predictable structure andconnectivity, which allows for efficient representation and processingin computers. Regularity also enables the use of varioussignal-processing techniques (e.g., for compression). A drawback ofregular meshes is their lack of flexibility in resolving shapes withhigh genus (an indicator of the number of handles or tunnels on a bodybounded by a closed surface) or detail at different levels. In contrast,irregular meshes are more flexible than regular meshes and can betterresolve complex geometric features, at the cost of greater complexity inrepresentation and processing.

Once a mesh model exists, it may be analyzed for surface quality (e.g.,fidelity to an original surface), representation efficiency, and othercriteria. Remeshing modifies the sampling and connectivity of ageometric mesh to generate a new mesh, for example, approximating thefirst mesh, refining the first mesh, or converting the first mesh toanother representation. Remeshing may involve using new or differentvertices for a mesh, and also may involve comparing samples from the newmesh to some original highly detailed surface. This process is calledresampling.

The number of vertices affects the complexity of a mesh. It alsocharacterizes the computational costs required to display or modify themesh. Hence, for computationally more expensive operations a mesh withfewer triangles would be preferred. Along that same vein, a highperformance computer can handle images and models of greater complexitysince they have more computational resources. Other factors such as meshregularity (discussed below) and mesh shape also may affectcomputational performance.

III. Regular Remeshing

Regular remeshing is a term describing methods whereby an irregular meshfor an object is approximated by a mesh with regular or semi-regularconnectivity. The simplicity of regularly remeshed representation hasmany benefits. One key benefit is that it eliminates the indirection andstorage of vertex indices and texture coordinates. (In a regular mesh,vertices and shape connectivity follow a predictable pattern. Graphicshardware is better able to assess which vertices it will need at anygiven time and have those loaded in memory.) This allows graphicshardware to perform rendering more efficiently by removing random memoryaccesses. Thus, memory performance improves.

Of various remeshing techniques, the geometry image (GIM) techniquecreates the most regular remeshed representation. The GIM technique wasintroduced by the article X. Gu, S. J. Gortler, and H. Hoppe, GeometryImages, SIGGRAPH 2002, pp. 355–361, 2002 (“Gu”). The constructiondescribed in the Gu article converts a surface into a topological diskusing a network of cuts, and parameterizes the resulting disk onto asquare domain. Using this parameterization, the surface geometry isresampled onto the pixels of an image. This technique has some benefits,including the fact that image compression can be directly applied to theremesh. However, the Gu approach of mapping an entire surface to asingle square has limitations. First, models having disconnectedcomponents require a separate geometry image for each component. Moreimportantly, complicated shapes with many extremities or topologicalhandles/tunnels have distorted parameterizations due to stretching andsquashing of the complicated shapes to flatten and fit them into theimage. On rendering, every sample within the geometry image is definedsince the model fills the image.

A less extreme approach than regular remeshing is to create a remeshwith the connectivity of a subdivided base mesh. Examples of thisapproach are sometimes called semi-regular remeshing, and includemethods described in the articles: (1) M. Eck, T. DeRose, T. Duchamp, H.Hoppe, M. Lounsberry, and W. Stuetzle, Multiresolution Analysis ofArbitrary Meshes, SIGGRAPH 1995, pp. 173–182, 1995 (“Eck”); (2) A. Lee,H. Moreton, and H. Hoppe, Displaced Subdivision Surfaces, SIGGRAPH 2000,pp. 85–94, 2000 (“Lee 1998”); (3) A. Lee, W. Sweldens, P. Schröder, L.Cowsar, and D. Dobkin, MAPS: Multiresolution Adaptive Parameterizationof Surfaces, SIGGRAPH 1998, pp. 95–104, 1998 (“Lee 2000”); (4) L.Kobbelt, J. Vorsatz, U. Labsik, and H. P. Seidel, A Shrink WrappingApproach to Remeshing Polygonal Surfaces, Eurographics 1999, pp.119–130, 1999 (“Kobbelt”); (5) I. Guskov, K. Vidimce, W. Sweldens, andP. Schröder, Normal Meshes, SIGGRAPH 2000, pp. 355–361, 2000 (“Guskov”);and (6) Z. J. Wood, M. Desbrun, P. Schröder, and D. Breen, Semi-regularMesh Extraction from Volumes, IEEE Visualization 2000, pp. 275–282, 2000(“Wood”). For example, a base version (i.e., a coarse, low-detailversion) of a mesh is created and sub-divided into base-level triangles,and additional details for the sub-divided base mesh are provided inadditional mesh layers. For these representations, special kinds ofcontinuous multiresolution basis functions can be derived that allowmultiresolution editing and compression.

These approaches provide a more flexible representation than pureregular remeshing, yet still have constraints that can negatively impactparameterization efficiency. In particular, each chart (here, a surfaceregion associated with a triangle of the base mesh) is effectivelyparameterized onto an equilateral triangle, which is then evenlysampled. Charts with non-triangular shapes are distorted by theparameterization. For example, charts that are long and skinny areinvariably sampled anisotropically (i.e., not exhibiting uniformityalong different axes), and are hence distorted. In addition, theseremesh techniques require that all charts, regardless of their size orinformation content, be allotted the same number of samples.

IV. Atlas Parameterization

Another framework for mesh processing and remeshing is atlasparameterization. Some atlas parameterization schemes map individualtriangles or pairs of triangles separately into a texture. For example,such techniques are described in the articles: (1) N. Carr, and J. Hartin Meshed Atlases for Real-time procedural solid texturing, ACMTransactions on Graphics. 21 (2), pp. 106–131, 2002 (“Carr”); and (2) P.Cignoni, C. Montani, C. Rocchini, and R. Scopigno in A General Methodfor Recovering Attribute Values on Simplified Meshes, IEEE Visualization1998, pp. 59–66, 1998 (“Cignoni”). Other, more general, chart-basedatlas constructions include those described in the articles: (1) J.Maillot, H Yahia, and A. Verroust, Interactive Texture Mapping, SIGGRAPH1993, pp. 27–34, 1993 (“Maillot”); (2) D. Piponi and G. D. Borshukov,Seamless Texture Mapping of Subdivision Surfaces by Model Pelting andTexture Blending, SIGGRAPH 2000, pp. 471–478, 1993 (“Piponi”); (3) P. V.Sander, J. Snyder, S. J. Gortler, and H. Hoppe, Texture MappingProgressive Meshes, SIGGRAPH 2001, pp. 409–416, 2001 (“Sander”); (4) B.Lévy, S. Petitjean, N. Ray, and J. Maillot, Least Squares Conformal Mapsfor Automatic Texture Atlas Generation, SIGGRAPH 2002, pp. 362–371, 2002(“Lévy”); (5) O. Sorkine, D. Cohen-Or, R. Goldenthal, and D. Lischinski,Bounded-distortion Piecewise Mesh Parameterization, IEEE Visualization2002, 2002 (“Sorkine”); and (6) A. Sheffer and J. C. Hart, Seamster:Inconspicuous Low-Distortion Texture Seam Layout, IEEE Visualization2002, 2002 (“Sheffer”).

The Maillot article describes partitioning a mesh into charts based onbucketing face normals. The parameterization method optimizes edgesprings of non-zero rest length.

The Piponi article describes manually cutting a subdivision surfaceusing a network of edges. This chart-base atlas constructionparameterizes the resulting single chart using a “pelting” analogy, bystretching out the surface boundary using a collection of springs.

The Sander article describes partitioning a mesh using greedy faceclustering. The construction parameterizes the resulting charts ontoconvex polygons using geometric stretch. The charts are packed into asquare using a greedy algorithm based on bounding rectangles. Areasbetween the convex polygons in the square are then filled with validvalues.

The Lévy article describes aligning chart boundaries with high-curvaturefeatures of the surface. After locating a set of seed faces farthestfrom sharp features, the technique grows charts about these seeds andmerges some of the resulting charts. Additionally, the Lévy articledescribes parameterizing each chart using least-squares conformal mapswith free boundaries and using a Tetris-like packing algorithm thatsearches for best fit over the horizon of pieces packed so far.

The Sorkine article describes simultaneously growing charts andparameterizing them. The chart growth stops when a distortion limit isreached or if self-overlap is detected. At this point, a new chart isstarted. The Sorkine parameterization uses a stretch-based metric thatpenalizes both undersampling and oversampling.

The Sheffer article describes cutting a surface into a single chart bycutting through high-distortion, less-visible surface regions.

A key drawback to these atlas constructions is that none addresses theproblem of inter-chart cracks in reconstructions due to resamplinggeometry images. Reconstructed signals are discontinuous across chartboundaries. Irregular chart outlines do not align with the samplinggrid; therefore, boundaries between neighboring charts are generallysampled differently. For geometry images, such signal discontinuitiescreate unacceptable cracks in the surface. In fact, even a few erroneouspixels become glaring artifacts.

Moreover, these atlas construction methods are greedy, thus theirtrade-off solutions between performance and final product may miss goodresults.

V. Model Decomposition

Others have investigated model decomposition outside of the context ofremeshing. For example, the article S. Shlafman, A. Tal, and S. Katz inMetamorphosis of Polyhedral Surfaces Using Decomposition, Eurographics2002, pp. 219–228, 2002 (“Shlafman”) describes a clustering-basedapproach to mesh partitioning. However, the Shlafman approach is usedfor morphing and not parameterization, therefore, its clusteringdistance metric does not account for overall chart planarity.

VI. Zippering

Outside of the context of remeshing, others have investigated how tohandle the problem of cracks between meshes during reconstruction. Forexample, the article G. Turk, and M. Levoy in Zippered Polygon Meshesfrom Range Images, SIGGRAPH 1994, pp. 311–318, 1994 (“Turk”) describesreconstructing watertight surfaces from scanned height meshes by“zippering” mesh boundaries together. In their implementation, thedesired surface is unknown, which makes their zippering technique achallenging operation. Therefore, there exists a need for a simple, yetrobust zippering technique.

Several others have considered the problem of reconstructing surfacesfrom contours, for example, in the article H. Fuchs, Z. Kedem, and S.Uselton, Optimal Surface Reconstruction from Planar Contours, Comm. ACM20, 1977 (“Fuchs”). The simplest case the Fuchs article describes isthat of building a surface ribbon that spans two parallel contourpolygons. This involves constructing a ribbon of new triangles along thegap between vertices instead of trying to unify the vertices across thegap.

Previous remeshing and chartification techniques do not provide anadequately efficient and simple 2D representation for arbitrary 3Dsurfaces, including those with extremities and handles/tunnels.Moreover, previous techniques do not address the problem of cracksbetween meshes during reconstruction. Finally, previous techniques leavemuch room for improvement for chartification solutions that are bothcomputationally practical and effective.

SUMMARY

Techniques and tools are presented herein for improving mesh processing.For example, multi-chart geometry images provide an efficient and simple2D representation for arbitrary 3D surfaces, including those withextremities or numerous handles and tunnels. Several chartificationsolutions provide computationally practical and effective ways topartition a 3D mesh into charts. And, zippering addresses the problem ofcracks between different areas of a reconstructed 3D mesh. The varioustechniques and tools can be used in combination or separately.

According to a first aspect, a tool such as a mesh processing toolgenerates a multi-chart geometry image. The tool partitions a mesh intomultiple charts with arbitrary, geometrically natural boundaries. Thetool then packs the charts into a single geometry image as irregularpolygons having arbitrary shapes based on the arbitrary, geometricallynatural boundaries. For example, the tool partitions the mesh byiteratively growing the charts and updating seeds for the charts. Afterthe partitioning process is complete, the tool packs the charts onto ageometry image by determining a scaling factor for the charts, selectingdimensions for the geometry image, and finding an appropriate placementfor each chart in the image. Even for 3D meshes with extremities ornumerous handles/tunnels, the multi-chart geometry image is an efficientand simple representation.

According to a second aspect, a tool such as a mesh processing toolzippers boundaries between charts for a mesh. The tool closes cracksbetween charts for a reconstruction of the mesh. For example, to closethe cracks, the tool identifies cut nodes in the charts, identifies cutpaths between adjacent charts, snaps together boundaries of adjacentcharts, and clusters together samples if necessary to seal remaininggaps. In this way, the tool provides a “watertight” surface for thereconstruction, preventing cracks that could otherwise appear due todiscretization of chart boundaries, for example, in a multi-chartgeometry image.

According to a third aspect, a tool such as a mesh processing toolperforms atlas chartification. The tool partitions a mesh surface into ageometrically natural set of charts for subsequent parameterization. Thepartitioning is based at least in part upon iterative clustering for thecharts. Thus, the tool partitions the mesh surface into charts in acomputationally practical and effective manner.

According to a fourth aspect, a tool such as a rendering tool performsrendering. The tool forms a reconstruction for a mesh from a geometryimage, handling one or more undefined samples in the geometry image. Thetool then creates an image from the reconstruction. For example, theundefined samples separate multiple charts in a multi-chart geometryimage.

Additional features and advantages of the invention will be madeapparent from the following detailed description of embodiments thatproceeds with reference to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a color illustration of an object model mesh partitioned intocharts.

FIG. 1B is a color illustration of a chart packing layout in an image.

FIG. 1C is a color illustration of a rendered reconstruction after theremeshing.

FIG. 2 is a block diagram of a mesh processing tool.

FIG. 3 is a flowchart showing a technique for producing a multi-chartgeometry image.

FIGS. 4A and 4B are color views of an object partitioned into chartsaccording to a prior art chartification technique.

FIG. 5 is a flowchart showing a technique for partitioning an objectinto charts.

FIGS. 6A and 6B are color views of an object partitioned into chartsaccording to the technique of FIG. 5.

FIG. 7A shows a color illustration of an object model divided intocharts.

FIG. 7B is a chart packing layout for the chartified object of FIG. 7Aaccording to a prior art packing algorithm.

FIG. 7C is a chart packing layout for the chartified object of FIG. 7Aaccording to the technique of FIG. 8.

FIG. 8 is a flowchart of a technique for packing charts into a layout.

FIG. 9 is a color illustration of criteria for evaluating chartplacements in an image.

FIGS. 10A and 10B are color illustrations of the packed charts beforeand after discretization, respectively.

FIGS. 11A–B are color illustrations of discretized packed charts beforeand after boundary rasterization, respectively.

FIGS. 12A–C are color illustrations of packed charts beforediscretization, after discretization but before local dilation, andafter local dilation.

FIG. 13 is a flowchart of a technique for zippering to close gapsbetween charts.

FIGS. 14A–D are color illustrations of a discretized images and areconstruction at various stages of the zippering process.

FIGS. 15A–C are color illustrations of boundary samples at differentstages of a unification and clustering process.

FIGS. 16A and 16B are color illustrations of reconstructions frommulti-chart geometry images before and after optimizations.

FIG. 17 is a table comparing results with single-chart and multi-chartgeometry images.

FIGS. 18A–D are color illustrations of chartified meshes, multi-chartgeometry images, and reconstructions for various models.

FIG. 19 is a set of color illustrations showing close up views of themodel and reconstruction of FIG. 18B.

FIG. 20 is a set of color illustrations showing three different texturemappings for a reconstruction from a multi-chart geometry image.

FIG. 21 is a set of color illustrations of a chartified mesh,multi-chart geometry image, and reconstruction for a model with severalconnected components.

FIGS. 22A–22C are color illustrations showing close up views of anoriginal irregular mesh, a reconstruction after semi-regular remeshing,and a reconstruction from a MCGIM.

FIG. 23 is a table comparing results with semi-regular remeshing andmulti-chart geometry images.

FIG. 24 is a chart relating multi-chart geometry image performance tothe number of charts.

FIG. 25 is a block diagram of an exemplary environment for implementingdescribed embodiments.

DETAILED DESCRIPTION

The following description is directed to techniques and tools forchartification, remeshing, and mesh reconstruction. These techniques andtools improve computer graphics performance in several ways. Describedtechniques and tools provide an efficient and simple 2D representationfor arbitrary 3D surfaces, including those with extremities andhandles/tunnels. Other techniques and tools address the problem ofcracks between meshes during reconstruction. Still other techniques andtools use chartification solutions that are both computationallypractical and effective.

Some embodiments involve multi-chart geometry images, which are a newrepresentation for arbitrary surfaces. A multi-chart geometry image(“MCGIM”) is a parameterization (e.g., atlas-based) created byresampling a surface onto a 2D grid such as a rectangle. A meshprocessing tool partitions the surface into a geometrically natural setof charts, parameterizes the charts individually onto irregularpolygons, and packs the polygons into an arbitrarily sized geometryimage. The geometrically natural set of charts may include charts withirregular boundaries, forming arbitrary shapes with concave or convexboundaries, or hybrids thereof. Whereas the GIM scheme described in theGu article maps the entire surface onto a single square, MCGIMsdescribed herein use an atlas construction to map a surface piecewiseonto charts of arbitrary shape, which are then packed into a geometryimage with those arbitrary shapes. This added flexibility reducesparameterization distortion and thus provides greater geometricfidelity, particularly for shapes with long extremities, high genus, ordisconnected components.

FIGS. 1A through 1C illustrate stages of an example of an MCGIM. FIG. 1Ashows a chartified original mesh 100, that is, a surface that has beenpartitioned into charts. Such an atlas parameterization reducesdistortion because the smaller charts are easier to flatten and becausetheir parameterized boundaries can assume more natural shapes.Low-distortion parameterizations distribute samples more uniformly overthe surface and therefore better capture surface signals. Each chart canbe allotted an appropriate number of “defined” samples in the geometryimage, separated by “undefined” samples. This representation can beviewed as piecewise regular since it is composed of sub-regions of aregular GIM.

FIG. 1B shows a geometry image 101 into which the charts have beenpacked. And, FIG. 1C shows a view 102 of the remesh after reconstructionfrom the charts in the image. MCGIMs retain a key advantage of theoriginal GIMs—rendering involves a simple scan of the samples in storedorder. And, MCGIMs overcome the problem of distortion present in GIMs,at the small expense of assigning some samples special “undefined”values.

In some embodiments, zippering algorithms create a watertight surface onmesh reconstructions. This prevents cracks, for example, by unifyingvertices across a gap between charts. Thus, a mesh processing toolunifies boundaries of discretized MCGIM charts to create a continuous,watertight model. (In contrast, traditional atlas constructions sufferfrom discontinuous reconstruction across chart boundaries, creatingunacceptable surface cracks.) The zippering algorithms are simple androbust, and typically use a given reference surface.

In some embodiments, new chartification schemes based on clusteringoptimizations are used. Based upon the general clustering optimizationwork of Lloyd and Max, these new chartification schemes create compact,roughly flat charts whose boundaries align with the model's creases.Generally speaking, these new chartification schemes are less greedythan previous methods, thus yielding better results.

Other embodiments involve still other improvements to existing atlasparameterization techniques. For example, applying the mesh optimizationframework described in the article Hoppe et al., Mesh Optimization,SIGGRAPH 1993, pp. 19–26 (“Hoppe”), to refit MCGIM samples to theoriginal surface, improves accuracy considerably. As another example,the “Tetris” packing algorithm described in the Lévy article is extendedto consider both chart rotations and overall domain dimensions.

Results indicate MCGIMs outperform both single-chart geometry images andsemi-regular remeshes on example models.

While the techniques and tools are described in places herein as part ofa single, integrated system, the techniques can be applied separately,potentially in combination with other techniques.

A mesh processing tool performs one or more of the various techniques.Alternatively, some other kind of tool performs one or more of thetechniques. Moreover, although the operations for these techniques aretypically described in a particular, sequential order for the sake ofpresentation, it should be understood that this manner of descriptionencompasses minor rearrangements in the order of operations, unless aparticular ordering is required. For example, operations describedsequentially may in some cases be rearranged or performed concurrently.Moreover, for the sake of simplicity, flowcharts typically do not showthe various ways in which particular techniques can be used inconjunction with other techniques.

I. Mesh Processing Tool

FIG. 2 shows an exemplary mesh processing tool 200 in which thetechniques described herein may be implemented. This system includeselements of software and/or hardware. As input, the tool 200 works witha geometric model 201, which can be a triangulated mesh or otherrepresentation of the geometry of the real or virtual object beingmodeled. The tool 200 produces MCGIMs that may be subsequently processedfor compression, decompression, rendering, or other operations.

In particular, FIG. 2 is a block diagram showing the various modules ofthe mesh processing tool 200. These modules include modules forchartification 210, polygon parameterization 220, chart packing 230,sampling 240, zippering 250, and mesh optimizations 260.

The relationships shown between modules within the mesh processing tool200 indicate the main flow of information; other relationships are notshown for the sake of simplicity. Depending on implementation, modulesof the tool can be added, omitted, split into multiple modules, combinedwith other modules, and/or replaced with like modules. Alternatively, atool with different modules and/or other configurations of modulesperforms the MCGIM, zippering, chartification, and/or other meshprocessing techniques described herein.

The chartification module 210 partitions the object model 201 intocharts and produces a set of charts. The number of charts can be staticor adaptive for the object model, defined for the tool or defined by theuser. For example, the chartification module 210 uses clustering inwhich clusters (here, charts) are determined by iteratively (1)partitioning elements (here, mesh faces) according to current clustermodels, and (2) refitting a model (here, a seed face for each chart) toeach cluster, until a termination condition is reached. Alternatively,the tool includes additional and/or different modules forchartification.

From a set of charts, the polygon parameterization module 220 produces aset of charts parameterized on a continuous domain. For example, thepolygon parameterization module 220 uses a hierarchical parameterizationalgorithm. Alternatively, the tool includes additional and/or differentmodules for chart parameterization.

The chart packing module 230 and sampling module 240 work to pack thecharts into an image and discretize the packed charts. For example, thechart packing module 230 evaluates packing orientations for differentscaling factors for the charts, different chart orientations, anddifferent image dimensions, and the sampling module 240 samples thechart boundaries and/or the packed charts. Alternatively, the toolincludes additional and/or different modules for chart packing and/orsampling.

The zippering module 250 accepts a set of charts packed into an imageand “zippers” boundary samples along discrete boundaries of charts thatadjoin on the 3D surface, which prevents cracks in the reconstructedmesh. For example, the zippering module 250 identifies “cut-nodes” inthe mesh, identifies “cut-paths,” snaps boundary samples to thecut-path, and unifies boundary samples. Alternatively, the tool includesadditional and/or different modules for zippering.

The mesh optimization module 260 optimizes samples in a “zippered” MCGIMto improve geometric fidelity to the original mesh. For example, themesh optimization module 260 uses mesh optimization techniques describedin the Hoppe article. Alternatively, the tool includes additional and/ordifferent modules for mesh optimization.

A rendering module 270 uses data received from the tool 200 to renderthe modeled object. In a desktop computer, the rendering module 270works with a display driver 280 to output images to an output device(e.g., to a monitor, projector, printer, or like device). The renderingmodule 270 is on the same computer as the various modules of the tool200. Alternatively, the modules of the tool 200 can be run on a separatecomputer, and the resulting data then transferred to the computer onwhich the rendering module 270 runs to display graphics images.

II. MCGIM Representation

Returning to FIGS. 1A through 1C, recall that a MCGIM is a rectangular2D grid whose entries store 3D points sampled from a given surface. AMCGIM consists of multiple, arbitrarily shaped charts packed togetherinto a grid. The charts may have irregular boundaries to form arbitraryshapes such as concave or convex polygons. FIG. 1B shows various colorcoded charts packed into a MCGIM 101, where the packed charts in FIG. 1Bare discretized (i.e., sampled) versions of the charts on the model 100in FIG. 1A (after parameterization to flatten the charts, etc.). Chartsare color-coded in the figures to help distinguish between charts, butthe colors do not indicate values for samples or pixels per se.

The 2D representation of the MCGIM stores coordinate (x, y, z) datavalues representing 3D points sampled from a reference surface. Likesome 3D representations, this 2D representation is compact becauseconnectivity information is implicit—how various vertices and facesconnect together is deduced from the stored coordinates. Hence, fewercomputer storage resources are required for the object. In addition, the2D representation makes it easy to associate colors and textures to theobject. For example, in some implementations, the 2D array of datavalues may further comprise color coordinates from a standard RGBcoordinate map. Or, two 2D images may be used to represent a singleobject—one image to represent the geometry of an object model and thesecond image to represent the colors associated with each sampled areaof the object. Texture may be similarly incorporated into the 2D imagemodel. Also, the relationship between resolution (i.e., samples) in the2D image and sampling resolution for the 3D mesh is relatively simpleand intuitive. Thus the 2D representation provides a flexible and simplebasis for representing a 3D model. Moreover, the 2D representation lendsitself to application of many current image filtering and compressiontechniques.

In most representations, there will be some samples in the image thatlie outside packed charts. These samples are called undefined samples.In FIG. 1B, the undefined samples are the yellowish samples separatingthe charts of various colors. Samples that lie within the charts(including chart boundaries) are called defined samples. Defined samplesfall into two main categories: boundary samples and interior samples. Aboundary sample is a sample with at least one undefined sample among itsfour immediate neighbors. An interior sample, therefore, is a samplewhere all of its four immediate neighbors are defined samples. Thedefined region of the image is specified either using a bitmask or bystoring NaN (Not-a-Number) into the coordinate values of undefinedsamples.

FIG. 1C illustrates a view 102 of a surface reconstructed from the MCGIM101 shown in FIG. 1B. In order to reconstruct a surface from the MCGIM101, polygon faces are formed from the data represented by 2D grids. Fora 2 by 2 quad of samples (i.e., a square of 4 points in the 2D grid),the number of defined samples is examined. If fewer than three of thesamples in the examined area are defined, no polygon is formed. Ifexactly three samples are defined, a triangle is formed. And if all foursamples are defined, two triangles are created, with the separatingdiagonal chosen to be the shorter of the two diagonals in the 2 by 2quad, according to 3D distance. Applying this method to the quads in theimage 101 creates a triangle mesh, which can be fed through an imagerendering engine or graphics pipeline for rendering. Notably, sometriangles may in fact be degenerate due to zippering (for example, whentwo vertices of a triangle come together and the triangle becomes aline, as discussed below); such degenerate triangles can be removedduring reconstruction, or, alternatively, they can be fed to thegraphics pipeline as extra points since they will have no effect onrendering.

For rendering, an image rendering engine computes surface normals ateach sample point of a MCGIM using one of a variety of techniques. Forexample, one technique for computing the surface normal at a pointinvolves calculating the normalized cross product of its neighbor'smathematically designated central differences. When an adjacent sampleis undefined, it is replaced by the current sample in the centraldifference. On a manifold mesh, opposite sample neighbors cannot both beundefined.

Ultimately, the goal of MCGIM construction is to create watertightmeshes, i.e. a 2-dimensional manifold where every edge is adjacent toexactly two faces and every vertex is adjacent to exactly one ring offaces. MCGIMs can represent meshes with several connected components, aswell as meshes with boundaries (where some edges are adjacent to onlyone face).

III. MCGIM Creation

FIG. 3 illustrates a technique 300 for creating a MCGIM. A tool such asthe mesh processing tool 200 shown in FIG. 2 performs the technique 300,and the ordering of stages in the technique 300 generally tracks themodules of the tool 200. Alternatively, another tool performs thetechnique 300.

The mesh processing tool creates a MCGIM by partitioning a mesh intomultiple charts 310 and parameterizing the charts 320. The tool packsthe charts into an image 330, iteratively evaluating different packinglayouts for the charts, and discretizes the charts 340. The tool zippersboundary samples 350 and performs further optimizations 360 to improvegeometric fidelity to the original mesh. FIG. 3 provides a map for thevarious stages of the technique 300, which are described in detailbelow. Alternatively, various stages of the technique 300 are separatelyor in various combinations performed in conjunction with other stages.

A. Mesh Chartification

As illustrated in FIG. 3, a first phase in producing a MCGIM ispartitioning the mesh into charts 310. Charts that have compactboundaries and are mostly flat yield better parameterization and packingresults. Moreover, they reduce the distortions in the image.

In one implementation, the chart merging approach of the Sander article(cited above) is used to chartify a mesh. The Sander approach applies agreedy face clustering algorithm to a model, which produces charts withirregular boundaries. To adjust the irregular boundaries, the Sanderarticle describes applying a straightening scheme to the charts'boundaries.

A problem with the Sander technique is that usually boundaries do notfollow the “creases” of the model, unless a high number of charts areused. Thus, to create a mesh in which relatively flat charts followcreases of the model, a high number of charts and resource expensivechartification computations are required. For instance, FIGS. 4A and 4Billustrate the results of a Sander chartification. FIG. 4A illustrates afront view 401 of a gargoyle object partitioned into charts. FIG. 4Bshows a top view 402 of the object partitioned into Sander charts. Uponcompletion of Sander chartification (including the straightening step),each of the gargoyle wings is composed of a single chart. If a singlechart stretches across folds/creases of the model, it is harder toparameterize that chart well for subsequent packing into an image, dueto the stretching and/or compaction of different parts of the chart asit is flattened. (In contrast, using different charts separated at thefolds/creases results in charts that are already flatter.)

This observation motivated the writers of the Lévy article to develop analgorithm that aligns chart boundaries with high-curvature features. TheLévy article describes estimating sharp features using local edgedihedral angles, identifying the seed faces farthest from sharpfeatures, and growing charts about these seeds. The Lévy approachimproved upon the Sander approach in some ways, but it still has itslimitations such as a feature detection requirement and lack of a goodgeneral optimization technique. Several techniques described herein usea more general optimization and chartification technique and do notrequire feature detection.

1. Chartification with Iterative Chart Growth and Seed Updating

Several techniques described herein employ an algorithm inspired byLloyd-Max quantization, which partitions sets of elements into a fixednumber of clusters. The basic strategy is to iteratively optimize theclusters using two main phases: (1) partitioning the elements accordingto current cluster models (chart growth phase) and (2) refitting a modelto each cluster (“seed computation phase). The strategy also usesbootstrapping and tests termination conditions.

In this context, elements are mesh faces, clusters are charts, and theclustering model tracks a seed face for each chart. FIG. 5 shows atechnique 500 for chartification of an object. A tool such as the meshprocessing tool 200 of FIG. 2 performs the technique 500. Alternatively,another tool performs the technique 500.

In the chart growth phase, the tool grows charts 510 according to thefollowing rule: given n seeds representing n charts, assign each meshface of the model to a chart. For example, the tool partitions elements(mesh faces) by growing all the charts simultaneously using a Dijkstrasearch algorithm on the “dual graph” of the mesh. A dual graph is agraph theory construct. In this case, the graph connects each pair ofadjacent mesh faces by an edge. Edges are assigned a cost thatencourages charts to be planar and to have compact boundaries.Specifically, the edge cost between a face F on a chart C and itsadjacent face F′ that is a candidate to join C is a measure of geometricdistance between the two faces and the difference in normal between F′and the chart normal N_(C):cost(F, F′)=(λ−(N _(C) −N _(F′)))(|P _(F′) −P _(F)|)  (1),where N_(C) is the normal of the chart containing F (the average normalof all faces already in the cluster), and P_(F′) and P_(F) are thecentroid points of F′ and F. The parameter λ regulates the relativeweight between the normal and geometric distance terms. In oneimplementation λ is 1. In other implementations, λ may have other valuessuch as ⅓, ½, 2, 3, 4, 10, 21, etc. The chart growth phase 410terminates when all faces have been assigned to charts. Alternatively,the tool uses additional and/or different search algorithms, distancemeasures, or graph constructs.

In the seed computation phase, the tool updates 520 the seed for eachchart, given a chartified mesh. For example, the tool updates the seedof each chart to be the “most interior” face within the chart. To findthis face, a Dijkstra search is performed on the dual graph of the mesh,but in the opposite direction. The search begins from all faces adjacentto a chart boundary. The edge cost is simply the geodesic distance:cost(F, F′)=|P _(F′) −P _(F)|  (2).

The last face reached within each chart becomes its new seed.Alternatively, the tool uses additional and/or different searchalgorithms, distance measures, or graph constructs.

The tool then evaluates 530 whether the chartification is finished, andif not, repeats the chart growth 510 and seed updating 520 phases. Forexample, the tool repeats the chart growth phase 410 and seedcomputation phase 420 until the new seeds are identical to those of theprevious iteration, thus indicating convergence. Alternatively, the tooluses additional and/or different termination conditions. Occasionally,seed values fail to converge because the seeds begin to cycle betweennearby faces (indicating nearly identical chartifications). In oneimplementation, such cycles are detected by checking the set of seedsagainst the ones obtained in previous iterations. If a cycle isdetected, any chartification in the cycle is acceptable.

The number n of charts can be static or adaptive for the object model.Thus, the same number of charts can be used no matter the effectiveness,or the number of charts can vary to trade-off the number vs. ease ofparameterization. In addition, the number n of charts can be initiallyset or defined by the tool or by the user. For example, in someimplementations, in order to bootstrap the chartification process, thetool assigns a random face as the first seed to begin generating charts.Once the initial seed has been selected, the tool runs successiveiterations of chart growth phase 410 and seed computation phase 420 withone small modification: at the end of chart growth phase 410, if adesired number of seeds has not been reached, a new seed is added. Thenew seed is set to be the last face that was assigned to a chart duringthe chart growth 410 phase. This tends to place the new seed far awayfrom the other seeds. Once the desired number of seeds has been reached,the chart growth phase 410 and seed computation phase 420 are repeatedwithout adding new seeds. The seeds may keep changing, but the number ofseeds stays the same. The tool also ensures the formation of properchart topology. For example, chart topology may require that charts aretopological disks and the formation of rings in charts and boundarylesscharts (closed charts) is not allowed. To enforce these constraints, thetool performs a simple local check on the chart boundary during chartgrowth to ensure proper chart topology. As a result of theseconstraints, however, there may be situations where a face cannot beassigned to any chart. When that happens, the first face that failedthis constraint check is added as a new seed. Then the new seed is grown(along with the other seeds) according to the chartification algorithm.For example, under the preceding constraints, a sphere cannot beaccurately chartified with only one chart. So, if one chart is requestedto represent the sphere's geometric data, more seeds are insertedautomatically, thus avoiding a closed chart. Alternatively, the toolensures the formation of proper chart topology according to additionaland/or different criteria.

2. Results

The chartification approach described herein produces better resultsthan previous techniques such as those described in the Sander and Lévyarticles. FIGS. 6A and 6B show an object partitioned into chartsaccording to the technique of FIG. 5. (For the sake of comparison, inFIGS. 4A, 4B, 6A, and 6B the object is a 12-chart gargoyle with the sameoriginal model.) FIG. 6A illustrates a front view 601 of the object, andFIG. 6B illustrates a top view 602 of the model. FIGS. 6A and 6B showthat the front and back of the ears and wings of the gargoyle model areeach assigned their own chart. Also, the chartification is much moresymmetric when compared to the Sander charts of FIGS. 4A and 4B. FIG. 6Bin particular illustrates the chart symmetry achieved by thechartification algorithm shown in FIG. 5.

To further evaluate the results of the two different chartificationapproaches, parameterize both chartified models and calculate theirstretch efficiencies. The stretch efficiency measures the distortion ofa given parameterization compared to the original model. The Sandermodel in FIGS. 4A and 4B has a stretch efficiency of just 80% (dueprimarily to the high stretch on the gargoyle wings). In contrast, thechartified model in FIGS. 6A and 6B has a near optimal stretchefficiency of 99%.

B. Chart Parameterization

As illustrated in FIG. 3, a second phase in producing a MCGIM isparameterizing the charts 320. To parameterize the charts, a meshprocessing tool parameterizes the charts into a continuous domain (latersampling depends on this parameterization). There are numeroustechniques for parameterization of charts.

In one implementation, the parameterization minimizes the L² geometricstretch metric disclosed in the Sander article. This metric penalizesundersampling and tends to result in samples uniformly distributed overthe mesh surface. Moreover, the stretch metric, as shown in the Sanderarticle, is a predictor of the reconstruction error under the assumptionof locally constant reconstruction. The tool allows chart boundaries tobe free during parameterization optimization, instead of fixing chartboundaries to a square like the Sander and Gu articles, which furtherreduces the stretch metric. In particular, the tool employs thehierarchical parameterization algorithm from the Sander article to solveminimization, improving quality and convergence speed. This hierarchicalalgorithm finds better minima than other techniques, and is orders ofmagnitude faster than uniresolution methods. Once all the charts areparameterized, they are scaled relative to each other based on theirstretch. Alternatively, the tool uses additional and/or differentparameterization techniques.

C. Chart Packing

As illustrated in FIG. 3, the next phase in producing a MCGIM is packingthe charts into an image 330, which involves iteratively evaluatingdifferent packing layouts for the charts. Packing a set of 2D chartsinto a single grid is a hard problem, typically requiring heuristicapproaches. There are several different techniques for packing chartsinto an image.

For example, a mesh processing tool uses an algorithm that is a simpleextension of the “Tetris” packing algorithm disclosed in the Lévyarticle. The Lévy scheme iteratively adds charts to a square image whilekeeping track of a “horizon” that bounds the charts packed so far. Foreach new chart to be inserted, the Lévy scheme locates the horizontalposition that minimizes the wasted vertical space between the currenthorizon and the new chart. FIG. 7A shows an object model 701 dividedinto 50 charts, and FIG. 7B shows a chart packing layout 702 for theobject 701 according to the Lévy scheme. The packing efficiency (thepercentage of the image made up of defined samples) is 58.0% for thelayout 702 in FIG. 7B.

Or, a mesh processing tool uses several enhancements to the Lévy packingscheme to dramatically improve its performance. For example, the toolconsiders 16 orientations for each chart, although more or lessorientations may be considered. And, the packing domain is an arbitraryrectangle, rather than a fixed-sized square. This is relevant sincecurrent graphics hardware can support textures with arbitrarydimensions. In one implementation, a user specifies a desired upperbound on the total number of MCGIM samples, and the tool selects anoptimal rectangle for the bound. In other implementations, the number ofsamples is automatically selected and the chart size is constrained insome other way.

FIG. 7C shows a chart packing layout 703 for the object 701 according tothe enhanced scheme, considering 16 orientations for charts and allowingarbitrary rectangle shapes for the image. The packing efficiency is75.6% for the layout 703 in FIG. 7C.—a tremendous improvement over thebasic Lévy chart packing algorithm.

FIG. 8 shows a technique 800 for packing charts into an image. A toolsuch as the mesh processing tool 200 of FIG. 2 performs the technique800. Alternatively, another tool performs the technique 800.

First, the tool computes a scaling factor 810 for the charts, which areparameterized in the continuous domain. The scaling factor determineshow the continuous charts map to the discrete image grid. That is, thescaling factor affects the number of samples that will be used later forthe charts in the image grid. In one implementation, the scaling factoris initially set based on the assumption that the tool will achieve apacking efficiency of 100% in the grid. If successful packing is notachieved, the scaling factor is successively reduced by 1% in subsequentiterations, until successful packing is achieved. For each scalingfactor, the charts are rasterized into a buffer to obtain their discreteshapes, where the shapes are later used for packing decisions.Rasterization is done to ensure chart topology, e.g., to prevent chartsfrom breaking into disconnected components. This concept is discussed inmore detail below. Alternatively, the tool uses additional and/ordifferent mechanisms to scale the charts.

Next, the tool computes the image dimensions 820 for the image where thecharts will be packed, varying the sides and overall size of the imagewithin applicable constraints. For example, for a given chart scale, thetool operates two nested loops, which consider MCGIM widths and heightswhose product (in terms of samples) is less than an upper bound on thetotal number of MCGIM samples. In one implementation, the aspect ratio(the ratio between the width of the picture and the height of thepicture) is limited to less than 4. In other implementations, the aspectratio has different limits.

The tool then determines chart placements 830 on the image, evaluatingdifferent locations and orientations for the charts in the image. Forexample, the tool examines various locations and orientations in orderto determine a chart placement strategy that minimizes wasted space inthe image. FIG. 9 shows several criteria for evaluating chart placementsin an image according to one implementation. The large red region iswasted space between the discretized upper horizon of the charts alreadyin the image (black line) and the discretized lower horizon of the chartbeing evaluated (green line). There is also wasted space within theintersection of the upper and lower envelope of the new chart (thesmaller red region in the upper-left), since the space is unused and adifferent rotation may avoid that wasted space. Alternatively, the tooluses additional and/or different placement strategies or criteria.

At the end of an iteration, the tool determines 840 whether the packinghas finished. If not, the tool continues with the next iteration of thetechnique 800. For example, the tool determines whether packing wassuccessfully achieved for a given scaling factor and, if not, continueswith an incrementally reduced scaling factor. Alternatively, the tooluses additional and/or different termination conditions.

D. Geometry Image Sampling

Once the desired scale and orientations of the charts within the imagehave been determined, the surface charts are discretized 340 into theimage. The pixels of the image take as values (x, y, z) positionssampled from the surface of the object model. FIG. 10 shows portions ofcontinuous charts 1001, which are discretized, resulting in the portionsof sampled charts 1002.

The set of defined image samples implicitly creates a triangulation, forexample, using the rules described earlier in connection with FIGS.1A–1C. This triangulation maintains the same topology as the originalsurface as long as the following two conditions have been satisfied: (1)the discretized charts are topological disks (the continuous surfacecharts are known to be topological disks, as discussed above) and (2)the charts are connected together without cracks (the explicit chartboundaries on the original mesh help to satisfy this condition). Thefirst condition is addressed below. The second condition is addressedbelow in the section entitled “Geometry image zippering.”

The tool rasterizes each chart's 3D triangles into the 2D image domainusing a scan conversion technique. During rasterization, the valuestored in the image domain sample is the corresponding (x, y, z)location from the 3D triangle. Thus, samples for packed charts arewritten to the 2D image. Unwritten samples (those not in charts) remainundefined.

This procedure does not guarantee that the rasterized charts will havedisk topology. For example, undersampling a narrow chart “peninsula”leads to broken charts, as illustrated in the discretized chart portions1101 of FIG. 11A. Hence, disk topology may be broken if only interiorrasterization is used.

To prevent charts from breaking into disconnected components, the toolrasterizes chart boundary edges to add connecting samples as shown inthe discretized packed charts 1102 of FIG. 11B. As necessary or for eachboundary edge, a 1-pixel wide rectangle centered along the boundary edge(called a boundary-polygon and shown in red) is constructed, andscan-converted to the texture domain. During this boundaryrasterization, undefined samples covered by the boundary-polygon arelabeled as defined and filled in with the closest geometric point on theboundary. Previously defined samples are not overwritten. This processhelps ensure that charts are properly connected.

Even with boundary rasterization, however, a discrete chart can have anon-manifold reconstruction at boundary samples, as illustrated by FIG.12B. Non-manifold reconstruction occurs when samples on a chart arepinched so that the discretized charts lose geometric fidelity. FIG. 12Aillustrates portions of packed charts to be discretized. In the smallersquare is highlighted a peninsula portion of one of the charts to bediscretized. In FIG. 12B, the now discretized chart has a single linepeninsula of defined samples, which will not produce any triangles onreconstruction. Reconstructions may also erroneously produce twotriangles touching at a single vertex. To correct these non-manifoldneighborhoods, the tool locally extends (dilates) the chart to includeadditional samples. FIG. 12C shows the discretized charts after localdilation. For each non-manifold sample p, the tool makes the samplemanifold by making “defined” a minimal number of its undefinedneighbors. The tool assigns these new samples the same position as p.This process is iterated until all samples are manifold.

In theory, there may be cases when this process fails, particularly whenthe requested geometry image area is small. For example, somediscretized charts could become annuli for coarse samplings. Forpractical image resolutions, however, these failures are not a problem.

Alternatively, the tool uses additional and/or different triangulationrules or discretization conditions.

E. Geometry Image Zippering

Referring again to FIG. 3, the next phase in producing a MCGIM iszippering the boundaries of charts within the image 350. Geometry imagezippering helps maintain an original surface topology. To prevent cracksin a reconstructed mesh, a mesh processing tool analyzes boundarysamples and “zippers” together discrete boundaries of charts that adjoinon the 3D surface.

FIG. 13 illustrates a technique 1300 for zippering along boundarysamples. FIGS. 14A–D illustrate a discretized image and reconstructionat various stages of the zippering process. A tool such as the meshprocessing tool 200 of FIG. 2 performs the technique 1300.Alternatively, another tool performs the technique 1300.

First, the tool identifies cut-nodes 1310 in the mesh. A cut-node is amesh vertex with more than two adjacent charts; in other words, it is achart corner. For every chart adjacent to the chart corner, the cut-nodeis mapped to a unique cut-node sample in the image domain. So, for agiven cut-node, different cut-node samples (for different charts) in theimage have the same (x, y, z) position value. Also, different cut-nodesamples for a given chart have different (x, y, z) position values. FIG.14A illustrates discretized charts 1401 in an image, with boundarysamples in black and cut-node samples in pink. In one implementation,the closest boundary sample in terms of 3D distance from the cut-node ischosen as the cut-node sample. Alternatively, the tool uses additionaland/or different criteria to identify cut-nodes or cut-node samples.

In addition, cut-node placement is constrained to preserve theclockwise-order of cut-paths (discussed below). In practice MCGIMresolution is typically large enough to map each cut-node to a distinctcut-node sample. If more than one cut-node sample does correspond to thesame cut-node, the cut-node samples are all assigned the same 3Dgeometric position corresponding to the cut-node.

Next, the tool identifies cut-paths 1320. FIG. 14B illustrates surfacecut-paths (in red lines) in a reconstruction 1402 of a rabbit meshmodel, before zippering. A cut-path is formed by connecting cut-nodes.Cut-nodes form the end points of a set of edges that make up thecut-path or boundary between two adjacent charts. A continuous surfacecut-path (e.g., a red line in FIG. 14B) in the mesh maps to two discretecut-paths (i.e., two sets of boundary samples) in the image domain.Boundary samples along these pairs of discrete cut-paths will be unifiedtoward the continuous cut-path.

To unify the boundary samples in opposing pairs of discrete cut-paths,the tool snaps chart boundaries into position 1330. FIG. 14C illustratesboundary samples from two adjacent charts snapped together to thesurface cut-paths in a reconstruction 1403. Each boundary sample in adiscrete cut-path is snapped to the closest point along the continuouscut-path, overwriting the value of that boundary sample in the discretecut-path with a new value. This narrows the gap between the charts. Butthe mesh may still not be watertight because of sampling differences, asshown in FIG. 14C. Two discrete cut-paths corresponding to the samecontinuous chart boundary generally contain different numbers ofsamples. If so, simple 1-to-1 unification along the two discretecut-paths is insufficient to seal the mesh because more than two samplesoccasionally may need to be unified.

Thus, as necessary, the tool seals boundaries 1340 between adjacentcharts. FIG. 14D illustrates boundary samples unified in areconstruction 1404 to form a watertight mesh. To seal the boundariescompletely, boundary samples are unified.

In one implementation, the tool performs a simple greedy algorithm toproduce high quality zippering. The algorithm uses a one-pass, lock-steptraversal of the two discrete cut-paths. FIGS. 15A–C show boundarysamples of two discrete cut-paths at different stages of this traversalprocess. In FIGS. 15A–C, the green bubbles represent unified clusters ofsamples, and the dashed lines represent two discrete cut-paths, one inpink and one in blue, with boundary samples along the cut path.

With reference to FIG. 15A, the greedy algorithm follows the cut-pathsand unifies boundary samples together. As illustrated, the tool hasalready snapped and unified two cut-node samples as described above(seen in the green bubble). As the traversal continues, the tool hasseveral options about how to proceed. For example, it may advance alongeither of the discrete cut-paths individually, as illustrated by thegreen arrow. In FIG. 15B a newly visited sample along the blue cut-pathis assigned to the geometric position of the current unified cluster.Notably, the greedy algorithm considers alternative alignments whenclustering samples. For example, referring back to FIG. 15A, the toolmay consider adding the second pink sample to the cluster (as shown bythe green line), but in this case determines that adding the second pinksample alters the updated sample too much. Or, the second pink sampleand the second blue sample may be clustered (as illustrated by thepurple line). Using the greedy algorithm, the tool considers thiscluster variation and selects it if it is the best one according to thealgorithm. Several techniques for choosing the best cluster alignmentmay work efficiently. Here, the zippering tool chooses the alignmentthat alters the updated samples by the least amount.

Instead of advancing along either of the discrete cut-pathsindividually, the tool may advance along both paths concurrently. FIG.15C illustrates the boundary samples with the tool advancing along bothof the discrete cut-paths concurrently. A new cluster of samples (in thesecond green bubble) may be created by the snap and unify techniquesdescribed above. The cluster of samples is assigned the geometricposition of one of the samples.

This process is repeated until all samples along the discrete cut-pathshave been unified with at least one sample on the other cut-path. Fordifferent alternative unification decisions, the tool typically selectsthe zipper advancement that alters the updated sample by the leastamount—a simple, greedy decision.

After unification, the mesh reconstructed from the discrete samples iswatertight. FIG. 14D shows a close-up image of the unified cut-paths inthe bunny reconstruction 1404.

Notably, in practice, when two adjacent boundary samples in the samediscrete cut-path are unified together, their reconstruction forms adegenerate triangle, which is in fact just a line. These degeneratefaces do not affect the manifold property of the mesh and can be removedduring reconstruction or harmlessly rendered as is. Even with texturemapping, degenerate faces do not introduce artifacts, since the texturesamples are unified just like the geometric ones.

Alternatively, a tool uses additional and/or different steps to zippertogether boundary samples across gaps between discretized boundaries ofadjacent charts, or works on charts other than charts in MCGIMs. Forexample, although the charts in an MCGIM are regular meshes (withirregular boundaries), the zippering techniques do not actually dependon the fact that the interior of a chart is regularly sampled. Thezippering techniques might be applied to multiple charts in differentimages, which are collectively used in reconstruction of a given meshsurface.

F. Geometry Image Optimization

After creating a zippered, watertight MCGIM, the mesh processing toolperforms mesh optimization 360 on the MCGIM. The MCGIM's defined samplesmay be optimized to improve geometric fidelity to the original mesh. Inother words, the (x, y, z) position values at pixels of the MCGIM may beadjusted, for example, to more accurately represent sharp features suchas creases and corners of the original object model. Moreover, imageoptimization may smooth and regularize the sample positions across thezippered chart boundaries.

For example, the mesh optimization technique described in the Hoppearticle is extended to optimize the mesh. In the Hoppe approach, boththe connectivity and vertex positions of the approximating mesh areoptimized to fit the reference surface. The fit accuracy is measured bysampling a dense set of points from the reference surface and minimizingthe squared distanced of those points from the approximating mesh. Inthis setting, the mesh connectivity is determined by the grid structureand the vertex positions. Each quad of the grid is split into twotriangles along the shorter diagonal for the quad. Thus, only the vertexpositions need to be optimized since the grid diagonals change asvertices are moved. Optimization occurs by using a non-linearconjugate-gradient solver (useful techniques for solving large-scalenon-linear optimization problems), which converges to reasonably goodoptimization answers, even though the fit measure lacks continuity dueto diagonal flipping. To keep the mesh from folding over during the fit,a mesh smoothness term is added to the energy function being minimized.This term is set to the L^(P) norm of the dihedral angles of the meshedges. In one implementation, setting p=4 produces good results. Inother implementations, the setting may be higher or lower than 4.

FIGS. 16A and 16B show examples of reconstructions from MCGIM modelsbefore and after optimization, with sharp features that are improvedusing geometry image optimization. FIG. 16A shows views 1601 and 1602 ofreconstructions from two MCGIMs before mesh optimizations. FIG. 16Bshows views 1611 and 1612 of reconstructions after mesh optimizations onthe MCGIMs.

Alternatively, the tool performs additional and/or different meshoptimization techniques.

IV. Rendering

In some embodiments, a tool such as a rendering tool accepts a MCGIM andprocesses it for rendering, handling undefined samples in the MCGIMduring the reconstruction process. Based upon the MCGIM, the renderingtool reconstructs part or all of a mesh for viewing from a givenperspective, distance, etc.

As described above, a MCGIM is a 2D array of samples that have valuesfor 3D points from a mesh. The charts packed into the MCGIM are made upof defined samples, and the charts are separated by undefined samples.The undefined samples are indicated by special invalid values in thegeometry image, by a separate bit mask indicating defined/undefinedstatus for the samples of the geometry image, or in some other manner.

In order to reconstruct all or part of the mesh surface from the MCGIM,the rendering tool traverses all or part of the samples of the MCGIM.For example, the rendering tool traverses the geometry image with achart-by-chart, raster scan, or other scan pattern. The regular samplingpattern for charts in the MCGIM simplifies rendering inasmuch as memoryaccess patterns for information such as vertices are more predictable,avoiding many memory indirections.

The rendering tool forms polygons such as triangles from the data in theMCGIM using reconstruction rules. For example, the rendering toolevaluates 2 by 2 groups of samples in the MCGIM. If fewer than three ofthe four samples have defined values, no triangle is formed. (Suchsamples typically represent a point or line that is part of anothertriangle and may be ignored during reconstruction.) If three of the foursamples have defined values, the tool forms one triangle. If all foursamples have defined values, the tool forms two triangles separated by adiagonal chosen from the two possible diagonals according to somecriterion. In one implementation, the criterion is shorter 3D distancefor the two possible diagonals. Alternatively, the tool uses additionaland/or different rules or criteria for forming triangles or otherpolygons during rendering from a MCGIM.

In alternative embodiments, a tool uses additional and/or differentsteps in rendering, or works with charts other than charts in MCGIMs.For example, although charts in an MCGIM are separated by undefinedsamples, the rendering described above could be applied to other sets ofmultiple charts (e.g., in different images) that have undefined samplesfor padding to regular image sizes, boundary shape preservation, etc.

V. Overall Results

A completely automatic pipeline has been run on a large number ofmodels. As a practical matter, the process takes approximately 1–2 hoursper model on the systems used. To give some idea of the relativecomputational complexity of the different tasks, if chartification takesno more than X minutes, parameterization no more than 3 times X minutes,sampling and zippering less than ⅕ times X minutes, and meshoptimization about 2 times X minutes. X varies according to the type ofsystem being used, for example on an exemplary test systems, asdisclosed herein, X was 5. Alternative systems may produce differentresults. The remainder of the time (6 times X to 18 times X) is neededfor the discrete optimization used for packing. Other packing schemesmay yield better results.

Two cost metrics to compare MCGIMs with other approaches are memory costand rendering cost, which may be defined as follows. Memory cost is thetotal number of samples in the domain grid (i.e., domain rectanglearea). Rendering cost is the number of samples that must be processed(i.e., defined samples). In the following cost metrics equation, packingefficiency is the ratio of defined samples to total number of samples.Thus,

rendering cost=memory cost*packing efficiency.

A measure of geometric accuracy is Peak Signal to Noise Ratio PSNR=20log₁₀(peak/d), where peak is the bounding box diagonal and d is thesymmetric rms Hausdorff error (geometric distance) between the originalmesh and the approximation. Similarly, the PSNR is measured for a normalsignal derived from the geometry. A large normal error suggestsparameterization artifacts like anisotropy.

A. Geometry Image Comparison

MCGIMs were created for a variety of models and compared them with GIMsaccording to the Gu article. To produce a fair comparison betweensingle-chart and multi-chart geometry images, geometry imageoptimization was also applied to the prior GIM results.

As shown in FIG. 17, a significantly better PSNR result is obtained byusing MCGIMs for the models in FIGS. 18A–18D. For the same memory cost(including storage cost for undefined samples), the geometry PSNR forthe MCGIMs for the horse and feline meshes was more than 10 dB higherthan with GIMs. To make a comparison for equivalent rendering cost, newMCGIMs were created with the same number of defined vertices as in therespective GIMs. The additional samples yield a further improvement ofabout 2 dB. For the Buddha model of FIG. 1A, the optimized GIM has aPSNR of 65.5 dB, whereas the equivalent-memory-cost MCGIM has a PSNR of71.7 dB.

FIG. 19 shows close up views of the model and reconstruction of FIG.18B. The top row is several close up views of the original mesh, themiddle row is several close up views of the reconstruction from a GIM,and the bottom row is several close up views of the reconstruction froma MCGIM.

FIG. 20 shows three different texture mappings using the implicit MCGIMparameterization. FIG. 20 shows a MCGIM model 2010 to be texture-mappedwith three images 2020, 2030, and 2040. The image 2040 is the samecoarse mesh, normal-mapped. The images 2020 and 2030 are texture-mappedusing a sampled, projected stripe texture at medium frequency and nearNyquist rate, respectively. Texture coordinates are implicit, and eachtriangle in the model is exactly associated with 9×9/2 texture samples.The resulting renderings are nearly seamless. One potential problem,however, is that some zippered triangles are longer than interior ones,resulting in local undersampling of the textures onto those longer,zippered triangles.

Unlike GIMs, MCGIMs can also effectively represent meshes with multiplecomponents. FIG. 21 shows a chartified mesh, MCGIM, and reconstructionfor a teapot mesh. The teapot object has four components: lid, spout,handle, and body. The chartified mesh 2110 has 30 charts. The resultingchartification and packing in the MCGIM 2120 are excellent (81.3%), andthe reconstructed mesh 2130 is produced with excellent results(PSNR=86.2 dB).

B. Semi-Regular Remeshing Comparison

Comparisons were also made with semi-regular remeshes created accordingto approaches described in the Guskov and Lee II articles. FIG. 22Ashows an example of views of an original irregular mesh. FIG. 22B showsviews of a semi-regular remesh. FIG. 22C shows views of a reconstructionfrom a MCGIM, with improved sampling uniformity compared to thesemi-regular remeshing.

For these comparisons, MCGIMs were created with rendering cost (i.e.,number of vertices) no larger than those of the semi-regular meshes,although for greater geometric fidelity the number could have beenhigher. FIG. 23 shows that the PSNR results with MCGIMs areapproximately 3 db higher than those of the semi-regular meshes,representing a 30% reduction in Hausdorff error (a binary image metric).

C. Selecting the Number of Charts

The number of charts used in an MCGIM influences geometric accuracy andrendering performance. FIG. 24 illustrates MCGIM accuracy as a functionof the number of charts. The chart indicates accuracy as the number ofcharts grows exponentially for the horse model, with 128² samples. MCGIMaccuracy exhibits a rough “hill” shape as the number of chartsincreases. Noise in the graph data comes from optimization proceduresinvolved in chartification, parameterization, zippering, and packing.Ultimately, however, the graph's shape comes from two countervailingtrends: MCGIMs with fewer charts use less gutter space and unify fewersamples; MCGIMs with more charts produce less parametric distortion andmore efficient packings. Thus, an optimal number of charts may exist fora given model, but good geometric accuracy is likely obtained over abroad range for the number of charts.

VI. Conclusion

Multi-chart geometry images provide a new surface representation thatextends the geometry images. MCGIMs have the same regular grid structureof GIMs, except that some of their image samples are undefined. Thispermits partitioning of the model into charts, and thus attains theflexibility needed to parameterize meshes of arbitrary complexity, highgenus, and multiple components with less distortion.

MCGIMs significantly exceed the geometric accuracy of GIMs forequivalent memory cost. This advantage improves even more when cost ofundefined samples is ignored, which is what happens duringreconstruction and rendering. MCGIMs also yield better geometricapproximations than those of semi-regular remeshing.

MCGIMs may be subjected to level-of-detail control and compression, andmay be rendered using graphics hardware, exploiting their simplicity infuture architectural designs.

VII. Computing Environment

The above described mesh processing tool 200 (FIG. 2) and meshprocessing and display techniques can be implemented on any of a varietyof computing devices and environments, including computers of variousform factors (personal, workstation, server, handheld, laptop, tablet,or other mobile), distributed computing networks, and Web services, as afew general examples. The graphical user interface with meshreconstruction display can be implemented in hardware circuitry, as wellas in mesh processing software 2580 executing within a computer or othercomputing environment, such as shown in FIG. 25.

FIG. 25 illustrates a generalized example of a suitable computingenvironment 2500 in which described techniques can be implemented. Thecomputing environment 2500 is not intended to suggest any limitation asto scope of use or functionality of the invention, as the presentinvention may be implemented in diverse general-purpose orspecial-purpose computing environments. By way of example and notlimitation, common implementations are in graphics cards and hardwareand/or software graphics engines for personal computers, console gamesystems, and specialized graphics environments.

With reference to FIG. 25, the computing environment 2500 includes atleast one processing unit 2510 and memory 2520. In FIG. 25, this mostbasic configuration 2530 is included within a dashed line. Theprocessing unit 2510 executes computer-executable instructions and maybe a real or a virtual processor. In a multi-processing system, multipleprocessing units execute computer-executable instructions to increaseprocessing power. The memory 2520 may be volatile memory (e.g.,registers, cache, RAM), non-volatile memory (e.g., ROM, EEPROM, flashmemory, etc.), or some combination of the two. The memory 2520 storessoftware 2580 implementing MCGIM processing.

A computing environment may have additional features. For example, thecomputing environment 2500 includes storage 2540, one or more inputdevices 2550, one or more output devices 2560, and one or morecommunication connections 2570. An interconnection mechanism (not shown)such as a bus, controller, or network interconnects the components ofthe computing environment 2500. Typically, operating system software(not shown) provides an operating environment for other softwareexecuting in the computing environment 2500, and coordinates activitiesof the components of the computing environment 2500.

The storage 2540 may be removable or non-removable, and includesmagnetic disks, magnetic tapes or cassettes, CD-ROMs, CD-RWs, DVDs, orany other medium which can be used to store information and which can beaccessed within the computing environment 2500. The storage 2540 storesinstructions for the software 2580 implementing MCGIM processing.

The input device(s) 2550 may be a touch input device such as a keyboard,mouse, pen, or trackball, a voice input device, a scanning device, oranother device that provides input to the computing environment 2500.For audio, the input device(s) 2550 may be a sound card or similardevice that accepts audio input in analog or digital form, or a CD-ROMreader that provides audio samples to the computing environment. Forsampling geometry, the input device(s) may be, for example, amechanical, optical, or other scanner. The output device(s) 2560 may bea display, printer, speaker, CD-writer, or another device that providesoutput from the computing environment 2500.

The communication connection(s) 2570 enable communication over acommunication medium to another computing entity. The communicationmedium conveys information such as computer-executable instructions,geometry, texture or other media information, or other data in amodulated data signal. A modulated data signal is a signal that has oneor more of its characteristics set or changed in such a manner as toencode information in the signal. By way of example, and not limitation,communication media include wired or wireless techniques implementedwith an electrical, optical, RF, infrared, acoustic, or other carrier.

The mesh processing techniques herein can be described in the generalcontext of computer-readable media. Computer-readable media are anyavailable media that can be accessed within a computing environment. Byway of example, and not limitation, with the computing environment 2500,computer-readable media include memory 2520, storage 2540, communicationmedia, and combinations of any of the above.

The techniques herein can be described in the general context ofcomputer-executable instructions, such as those included in programmodules, being executed in a computing environment on a target real orvirtual processor. Generally, program modules include routines,programs, libraries, objects, classes, components, data structures, etc.that perform particular tasks or implement particular abstract datatypes. The functionality of the program modules may be combined or splitbetween program modules as desired in various embodiments.Computer-executable instructions for program modules may be executedwithin a local or distributed computing environment.

For the sake of presentation, the detailed description uses terms like“determine,” “generate,” “adjust,” and “apply” to describe computeroperations in a computing environment. These terms are high-levelabstractions for operations performed by a computer, and should not beconfused with acts performed by a human being. The actual computeroperations corresponding to these terms vary depending onimplementation.

In view of the many possible embodiments to which the principles of ourinvention may be applied, we claim as our invention all such embodimentsas may come within the scope and spirit of the following claims andequivalents thereto.

1. A computer-readable memory storing computer-executable instructionsfor causing a computer system programmed thereby to perform a method ofzippering boundaries between charts for a mesh, the method comprising:identifying a plurality of charts for a mesh having a reference surface;and closing one or more cracks between the plurality of charts for areconstruction of the mesh, wherein the closing is based at least inpart on the reference surface; wherein the closing comprises:identifying a plurality of cut-nodes in the plurality of charts, whereeach of the plurality of cut-nodes comprise a mesh vertex with at leastthree adjacent charts; and mapping each cut-node to a closest boundarysample in the plurality of charts in terms of 3D distance from thecut-node.
 2. The computer-readable memory of claim 1 wherein theplurality of charts are in a multi-chart geometry image.
 3. Thecomputer-readable memory of claim 1 wherein at least one of theplurality of charts has an irregular boundary but not a regularlysampled interior.
 4. A computer-readable memory storingcomputer-executable instructions for causing a computer systemprogrammed thereby to perform a method of zippering boundaries betweencharts for a mesh, the method comprising: identifying a plurality ofcharts for a mesh having a reference surface; and closing one or morecracks between the plurality of charts for a reconstruction of the mesh,wherein the closing is based at least in part on the reference surface;wherein the closing comprises: identifying one or more cut paths betweenadjacent charts of the plurality of charts, wherein each of the one ormore cut paths comprises at least a continuous cut path and a pair ofdiscrete cut paths for the respective adjacent charts; and snappingtogether boundaries of the adjacent charts on the pair of discrete cutpaths.
 5. The computer-readable memory of claim 4 wherein the pluralityof charts are in a multi-chart geometry image.
 6. The computer-readablememory of claim 4 wherein at least one of the plurality of charts has anirregular boundary but not a regularly sampled interior.
 7. Acomputer-readable memory storing computer-executable instructions forcausing a computer system programmed thereby to perform a method ofzippering boundaries between charts for a mesh, the method comprising:identifying a plurality of charts for a mesh having a reference surface;and closing one or more cracks between the plurality of charts for areconstruction of the mesh, wherein the closing is based at least inpart on the reference surface; wherein the closing comprises: clusteringeach of plural subsets of boundary samples in adjacent charts of theplurality of charts to seal gaps.
 8. The computer-readable memory ofclaim 7 wherein the clustering comprises: visiting plural boundarysamples along two discrete cut paths; and assigning at least one of theplural boundary samples to a cluster according to a selection mechanism.9. The computer-readable memory of claim 7 wherein the plurality ofcharts are in a multi-chart geometry image.
 10. The computer-readablememory of claim 7 wherein at least one of the plurality of charts has anirregular boundary but not a regularly sampled interior.
 11. Acomputer-implemented method of rendering comprising: receiving ageometry image; and forming a reconstruction for a mesh based at leastin part upon the geometry image, including handling one or moreundefined samples in the geometry image; wherein the forming comprisestraversing the geometry image and forming triangles of thereconstruction according to plural rules, and wherein the plural rulesinclude: forming no triangle if less than three samples are defined in a2-by-2 set of samples, forming one triangle if three samples are definedin the 2-by-2 set of samples, and forming two triangles if four samplesare defined, wherein a diagonal separates the two triangles in the2-by-2 set.
 12. The method of claim 11 wherein the geometry imageincludes plural charts having regularly sampled interiors, therebyreducing memory accesses during the forming.
 13. The method of claim 11wherein the geometry image includes plural defined samples for pluralcharts and plural undefined samples separating the plural charts. 14.The method of claim 11 wherein the geometry image is rectangular. 15.The method of claim 11 wherein the reconstruction is a partialreconstruction of the mesh.
 16. A computer-readable memory storingcomputer-executable instructions for causing a computer systemprogrammed thereby to perform the method of claim
 11. 17. A graphicsdevice including one or more of hardware and software logic forperforming the method of claim 11.